Lesson Notes By Weeks and Term - Senior Secondary School 1

SUBJECT: MATHEMATICS

CLASS:  SS 1

DATE:

TERM: 1st TERM

REFERENCE BOOKS

• New General Mathematics SSS 1     M.F. Macrae et al 

 
WEEK THREE

TOPIC: BINARY NUMBERS (BASE 2 NUMBERS)

• Addition in base 2

• Subtraction in base 2

• Multiplication & Division in base 2

ADDITION IN BASE TWO

We can add binary numbers in the same way as we separate with ordinary base 10 numbers. 

The identities to remember are:-

0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10, 1 + 1 + 1 = 11, 1 + 1 + 1 + 1 = 100

Worked Examples

Example 1

Simplify the following

1. 1110 + 1001           2.      1111 + 1101 + 101

Solutions:

1. 1110

        +    1001

10111

2. 1111

         + 1101

  101

100001

Note: 11 take 1 carry 1 

         10 take 0 carry1

         100 take 0 carry 10

Example 2 

• 11011two + 1111two

• 10011 + 1110 

• 110111 + 11011 + 10111

Solution 

1. 11011two + 1111two

1 1 0 1 1

   1 1 1 1 

        1 0 1 0 1 0 

1. 10011 + 1110

1 0 0 1 1 

   1 1 1 0 

        1 0 0 0 0 1 

1. 110111 + 11011 + 10111

1 1 0 1 1 1 

   1 1 0 1 1

   1 0 1 1 1

                1 1 0 1 0 0 1

EVALUATION

1. Simplify the following; 1001 + 101 + 1111
2. 10101 + 111

SUBTRACTION IN BASE TWO

The identities to remember on subtraction are: 0 - 0 = 0, 1 - 0 = 1, 10 - 1 = 1, 11 - 1 = 10, 100 - 1 = 11

Worked Examples

Simplify the following:-

(a)    1110 - 1001    (b) 101010 - 111

Solutions:

(a)     1110

        -   1001

      101

(b)       101010

         -       111

          1110

Example 2

1. 1001two – 111two
2. 10001 – 1111 
3. 11010two – 1111two

Solution 

1. 1001two – 111two

1 0 0 1 

   1 1 1

      1 0

1. 10001 – 1111 

1 0 0 0 1 

   1 1 1 1 

         1 02

1. 11010two – 1111two

1 1 0 1 0

   1 1 1 1

      1 0 1 1

MULTIPLICATION AND DIVISION IN BASE TWO

In multiplication, 0 x 0 = 0, 1 x 0 = 0, 1 x 1 = 1.

When there is long multiplication of binary numbers, the principle of addition can be used to derive the answer. Under division, the principle of subtraction can be used.

Worked Examples:

1. 1110 x 111 2.    110 ÷ 10

Solution:

1. 1110                                 2.                   11

x   110                                               10   110

            0000                                                       10

          1110                                                           10

1110      10

1010100                                                      00

Example 2 

1. 101011 X 110 
2. 11101 X 111

Solution 

• 101011 X 110 

The working is shown below without explanation 

1 0 1 0 1 1

         1 1 0

0 0 0 0 0 0

1 1 0 1 0 1 

  1 1 0 1 0 1

   1 0 0 0 0 0 0 1 0

• 11101 X 111

1 1 1 0 1 

      1 1 1

1 1 1 0 1

  1 1 1 0 1

          1 1 1 0 1

   1 1 0 0 1 0 1 1

Example 3 

1. 101010 111 (base two)

110

    111    101010

        111

          111

          111

            00

            00

1. Divide 1010.01two by 11two giving your answer to 3 places after the binary point. 

11.011

            11    1010.010

                      -   11

                  100

                  -11

                    101

                    -11

                        100

                          11

                    1

EVALUATION

1. Evaluate    10111÷110   
2. Evaluate 10001 x 11      
3. Evaluate 10001 - 1110

GENERAL EVALUATION

1. Evaluate 111101 x 111          
2. Evaluate 40205 ÷ 115 
3. 11001 + 1111     
4. 1101 – 111
5. 1 1 1 1

    1 1 0

READING ASSIGNMENT

Essential Mathematics for SS 1 pages 54 – 55 

WEEKEND ASSIGNMENT

1. Express 3426 as a number in base 10.     (a) 342        (b) 3420       (c) 134
2. Change the number 10010 to base 10     (a) 18       (b) 34           (c) 40
3. Express in base two the square of 11     (a) 1001   (b) 1010       (c)  1011
4. Find the value of (101)2 in base two      (a) 1010   (b) 1111       (c)  1001
5. Multiply 1000012 by 11         (a) 1001   (b) 1100011 (c)  10111

THEORY

• Convert the following to binary number 

1. 10ten
2. (10ten)2

1. Calculate 1102 x (10112 + 10012 – 1012)
2. Multiply 345 by 225.